Let's not try and fix too many things at once. I don't think changing the chamber will help.
You're still struggling with a B3, with a No2 reed. Is the struggling still the volume change?
Do you change your embouchure/blowing to get the higher notes with the octave key pressed?
First thought is that you need to get the sax checked. Before you go completely crazy. Lots of things stem from leaks.
You should be pretty much able to press the octave key and the sax should shift up an octave, with only a little change in the way you blow, possibly a touch more lip, but not a lot.
Assuming the sax is OK, a slightly harder reed may help. It's worth going up half a step at a time. And stick to the same type of reed - different brands have differing strengths for the same number, and within the brand different reed types blow differently. Given what you've said I wouldn't change mouthpiece at the moment. Check the sax... And if you can, get teh person who checks it to play it while you listen. Then you know what to expect.